#8745795

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#56439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,27)(20,26)(21,24)(22,28)(23,25)(29,31)(30,32)(33,34,39,37)(35,43,46,40)(36,41,45,47)(38,42,44,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,32,26)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(2,3)(7,9)(8,10)(15,16)(18,25)(20,31)(24,27)(28,30)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,26)(19,28)(21,30)(22,27)(23,31)(24,32)(25,29)(33,36)(34,41)(35,44)(37,47)(38,46)(39,45)(40,42)(43,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,32,26)(33,37,39,34)(35,43,46,40)(36,47,45,41)(38,42,44,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22)(18,28)(19,29)(20,24)(21,26)(23,32)(25,30)(27,31)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8652991);